This is problem 6.11 of Rotman Homological Algebra. I have solved the problem. However, I am wondering whether the induced long exact sequence is some long exact sequence between homology of complexes.
Consider the following diagram with both rows exact in some abelian category.
$\dots \to A_n\to B_n\to C_n\to A_{n-1}\to B_{n-1}\to C_{n-1}\to\dots$
$\dots \to A'_n\to B'_n\to C'_n\to A'_{n-1}\to B'_{n-1}\to C'_{n-1}\to\dots$
There are morphisms $f_n:A_n\to A'_n,g_n:B_n\to B'_n, h_n:C_n\to C'_n$ for each $n\in Z$, $A_n\to A_{n-1}$ is the connecting morphism and $h_n$ are isomorphism for all $n\in Z$.
There will be induced exact sequence $\dots\to A_n\to A'_n\oplus B_n\to B'_n\to A_{n-1}\to A'_{n-1}\oplus B_{n-1}\to B'_{n-1}\dots$. The proof of exact sequence is trivial.
Normally given exact sequence $0\to C'\to C\to C''\to 0$ where $C',C,C''\in Compx(\mathcal{A})$, the category of complexes over some abelian category, I have induced long exact sequence from homology. So given morphism from two exact sequences of complexes, I have induced morphism for homology of each exact sequence and this becomes long exact sequence(ladder diagram) in some appropriate category.
Is it possible to regard Barrat Whitehead induced long exact sequence as long exact sequence of some category of complex over abelian category?